See Figure REF  for a comparison of reduced and \(m\) 'th iterate maps. If the number of iterates \(m \in \mathbb {N}_+\)  is comparable to \(\varepsilon ^{-1}\) , then iterates of the \(m\) 'th iterate map are generically separated by \(O(m \varepsilon ) = O(1)-\) distances. In essence, the \(\varepsilon -\) dependence cancels if one `speeds up' the map by considering only every \(m\) 'th iterate with \(m = \varepsilon ^{-1} \tilde{m}\)  for some \(\tilde{m} > 0\) . This has a similar effect to the time rescaling \(t = \varepsilon ^{-1} \tau \)  in fast-slow ODEs. In fact the \(m\) 'th iterate map (REF ) relates directly to the ODE reduced problem \(z^{\prime } = \Pi _{\mathcal {N}}^{S_n} G(z,0)\)  associated to general fast-slow systems (REF ) (see [1]}), via the Euler discretization of the latter. Specifically, the map (REF ) can be rewritten as
\(\frac{{\bar{z}}^m - z}{\tilde{m}} = \Pi _{\mathcal {N}}^{S_n} G(z,0) ,\) 
